Related papers: Ensemble equivalence for dense graphs
We analyse the size of an independent set in a random graph on $n$ vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent…
We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show…
In a network cliques are fully connected subgraphs that reveal which are the tight communities present in it. Cliques of size c>3 are present in random Erdos and Renyi graphs only in the limit of diverging average connectivity. Starting…
We study the behaviour of random labelled and unlabelled cographs with n vertices as n tends to infinity. Our main result is a novel probabilistic limit in the space of graphons.
We present an analytical approach for describing spectrally constrained maximum entropy ensembles of finitely connected regular loopy graphs, valid in the regime of weak loop-loop interactions. We derive an expression for the leading two…
Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…
We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with…
Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple…
Yin, Rinaldo, and Fadnavis classified the extremal behavior of the edge-triangle exponential random graph model by first taking the network size to infinity, then the parameters diverging to infinity along straight lines. Lubetzky and Zhao…
For studying the thermodynamic properties of systems using statistical mechanics we propose an ensemble that lies in between the familiar canonical and microcanonical ensembles. From a comparative study of these ensembles we conclude that…
We study edge-isoperimetric inequalities in chamber graphs of affine hyperplane arrangements. Our approach is topological: to a set of chambers we associate its thickening in Euclidean space and estimate its edge boundary through the…
This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are…
In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky…
In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform…
We show that the random number $T_n$ of triangles in a random graph on $n$ vertices, with a strict constraint on the total number of edges, admits an expansion $T_n = an^3 + bn^2 + F_n$, where $a$ and $b$ are numbers, with the mean $\langle…
We study correlation decay for the maximum weight matching problem on sparse graphs with i.i.d. edge weights. We show exponential decay of correlations when the underlying graphs are locally tree-like with uniformly bounded degree and the…
The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal…
Graph neural networks (GNNs) continue to achieve state-of-the-art performance on many graph learning tasks, but rely on the assumption that a given graph is a sufficient approximation of the true neighborhood structure. When a system…
We generate new mathematical tools with which to quantify the macroscopic topological structure of large directed networks. This is achieved via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random…
Ensemble models often achieve higher accuracy than single learners, but their ability to maintain small generalization gaps is not always well understood. This study examines how ensembles balance accuracy and overfitting across four…